Scheduling Bipartite Tournaments to Minimize Total Travel Distance

In many professional sports leagues, teams from opposing leagues/conferences compete against one another, playing inter-league games. This is an example of a bipartite tournament. In this paper, we consider the problem of reducing the total travel distance of bipartite tournaments, by analyzing inter-league scheduling from the perspective of discrete optimization. This research has natural applications to sports scheduling, especially for leagues such as the National Basketball Association (NBA) where teams must travel long distances across North America to play all their games, thus consuming much time, money, and greenhouse gas emissions. We introduce the Bipartite Traveling Tournament Problem (BTTP), the inter-league variant of the well-studied Traveling Tournament Problem. We prove that the 2n-team BTTP is NP-complete, but for small values of n, a distance-optimal inter-league schedule can be generated from an algorithm based on minimum-weight 4-cycle-covers. We apply our theoretical results to the 12-team Nippon Professional Baseball (NPB) league in Japan, producing a provably-optimal schedule requiring 42950 kilometres of total team travel, a 16% reduction compared to the actual distance traveled by these teams during the 2010 NPB season. We also develop a nearly-optimal inter-league tournament for the 30-team NBA league, just 3.8% higher than the trivial theoretical lower bound

Problem collection from the IML programme: Graphs, Hypergraphs, and Computing

This collection of problems and conjectures is based on a subset of the open problems from the seminar series and the problem sessions of the Institut Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem contributor has provided a write up of their proposed problem and the collection has been edited by Klas Markstr\"om.Comment: This problem collection is published as part of the IML preprint series for the research programme and also available there http://www.mittag-leffler.se/research-programs/preprint-series?course_id=4401. arXiv admin note: text overlap with arXiv:1403.5975, arXiv:0706.4101 by other author

Proximity and Remoteness in Directed and Undirected Graphs

Let $D$ be a strongly connected digraph. The average distance $\bar{\sigma}(v)$ of a vertex $v$ of $D$ is the arithmetic mean of the distances from $v$ to all other vertices of $D$. The remoteness $\rho(D)$ and proximity $\pi(D)$ of $D$ are the maximum and the minimum of the average distances of the vertices of $D$, respectively. We obtain sharp upper and lower bounds on $\pi(D)$ and $\rho(D)$ as a function of the order $n$ of $D$ and describe the extreme digraphs for all the bounds. We also obtain such bounds for strong tournaments. We show that for a strong tournament $T$, we have $\pi(T)=\rho(T)$ if and only if $T$ is regular. Due to this result, one may conjecture that every strong digraph $D$ with $\pi(D)=\rho(D)$ is regular. We present an infinite family of non-regular strong digraphs $D$ such that $\pi(D)=\rho(D).$ We describe such a family for undirected graphs as well

An Average Case NP-Complete Graph Coloring Problem

NP-complete problems should be hard on some instances but those may be extremely rare. On generic instances many such problems, especially related to random graphs, have been proven easy. We show the intractability of random instances of a graph coloring problem: this graph problem is hard on average unless all NP problem under all samplable (i.e., generatable in polynomial time) distributions are easy. Worst case reductions use special gadgets and typically map instances into a negligible fraction of possible outputs. Ours must output nearly random graphs and avoid any super-polynomial distortion of probabilities.Comment: 15 page

Skill Rating for Generative Models

We explore a new way to evaluate generative models using insights from evaluation of competitive games between human players. We show experimentally that tournaments between generators and discriminators provide an effective way to evaluate generative models. We introduce two methods for summarizing tournament outcomes: tournament win rate and skill rating. Evaluations are useful in different contexts, including monitoring the progress of a single model as it learns during the training process, and comparing the capabilities of two different fully trained models. We show that a tournament consisting of a single model playing against past and future versions of itself produces a useful measure of training progress. A tournament containing multiple separate models (using different seeds, hyperparameters, and architectures) provides a useful relative comparison between different trained GANs. Tournament-based rating methods are conceptually distinct from numerous previous categories of approaches to evaluation of generative models, and have complementary advantages and disadvantages

Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms

In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time

Efficiently sampling the realizations of irregular, but linearly bounded bipartite and directed degree sequences

Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. Several results were proved on rapidly mixing Markov chains on regular simple, on regular directed, on half-regular directed and on half-regular bipartite degree sequences. In this paper, the main result is the following: Let $U$ and $V$ be disjoint finite sets, and let $0 < c_1 \le c_2 < |U|$ and $0 < d_1 \le d_2 < |V|$ be integers. Furthermore, assume that the bipartite degree sequence on $U \cup V$ satisfies $c_1 \le d(v) \le c_2\ : \ \forall v\in V$ and $d_1 \le d(u) \le d_2 \ : \ \forall u\in U$. Finally assume that$(c_2-c_1 -1)(d_2 -d_1 -1) < 1 + \max \{c_1(|V| -d_2), d_1(|V|- c_2) \}$. Then the swap Markov chain on this bipartite degree sequence is rapidly mixing. The technique applies on directed degree sequences as well, with very similar parameter values. These results are germane to the recent results of Greenhill and Sfragara about fast mixing MCMC processes on simple and directed degree sequences, where the maximum degrees are$O(\sqrt{\# \mbox{ of edges}})$. The results are somewhat comparable on directed degree sequences: while the GS results are better applicable for degree sequences developed under some scale-free random process, our new results are better fitted to degree sequences developed under the Erd\H{o}s -- R\'enyi model. For example our results cover all regular degree sequences, the GS model is not applicable when the average degree is$ > n/16.$Comment: 20 pages. arXiv admin note: text overlap with arXiv:1301.752

A Fast MCMC for the Uniform Sampling of Binary Matrices with Fixed Margins

Uniform sampling of binary matrix with fixed margins is an important and difficult problem in statistics, computer science, ecology and so on. The well-known swap algorithm would be inefficient when the size of the matrix becomes large or when the matrix is too sparse/dense. Here we propose the Rectangle Loop algorithm, a Markov chain Monte Carlo algorithm to sample binary matrices with fixed margins uniformly. Theoretically the Rectangle Loop algorithm is better than the swap algorithm in Peskun's order. Empirically studies also demonstrates the Rectangle Loop algorithm is remarkablely more efficient than the swap algorithm

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application

Generating Approximate Solutions to the TTP using a Linear Distance Relaxation

In some domestic professional sports leagues, the home stadiums are located in cities connected by a common train line running in one direction. For these instances, we can incorporate this geographical information to determine optimal or nearly-optimal solutions to the n-team Traveling Tournament Problem (TTP), an NP-hard sports scheduling problem whose solution is a double round-robin tournament schedule that minimizes the sum total of distances traveled by all n teams. We introduce the Linear Distance Traveling Tournament Problem (LD-TTP), and solve it for n=4 and n=6, generating the complete set of possible solutions through elementary combinatorial techniques. For larger n, we propose a novel "expander construction" that generates an approximate solution to the LD-TTP. For n congruent to 4 modulo 6, we show that our expander construction produces a feasible double round-robin tournament schedule whose total distance is guaranteed to be no worse than 4/3 times the optimal solution, regardless of where the n teams are located. This 4/3-approximation for the LD-TTP is stronger than the currently best-known ratio of 5/3 + epsilon for the general TTP. We conclude the paper by applying this linear distance relaxation to general (non-linear) n-team TTP instances, where we develop fast approximate solutions by simply "assuming" the n teams lie on a straight line and solving the modified problem. We show that this technique surprisingly generates the distance-optimal tournament on all benchmark sets on 6 teams, as well as close-to-optimal schedules for larger n, even when the teams are located around a circle or positioned in three-dimensional space